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Question 2 of 22
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Q1
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Q2
$5 x$ points
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$2 x$ points
Q4
(a) $\lim _{x \rightarrow 0} \frac{x}{\sin 8 x}=$
$$
\frac{1}{8}
$$
Devely
$0118-$
$0=0$
,
C
64
2x points
Q5
$2 x$ points
Q6
(b) $\lim _{x \rightarrow 0} \frac{x}{\cos 8 x}=$
0
2x points
(c) $\lim _{x \rightarrow 0} \frac{x}{\tan 8 x}=\frac{1}{8}$

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∻Solution by Steps∻ ‖ step 1 ⋮ To find the derivative of the function $ f(x) = \frac{e^2}{x^3 + 1} $, we apply the quotient rule, which states that if $ f(x) = \frac{g(x)}{h(x)} $, then $ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} $. Here, $ g(x) = e^2 $ and $ h(x) = x^3 + 1 $. ‖ ‖ step 2 ⋮ The derivatives are $ g'(x) = 0 $ (since $ e^2 $ is a constant) and $ h'(x) = 3x^2 $. ‖ ‖ step 3 ⋮ Substituting into the quotient rule gives $ f'(x) = \frac{0 \cdot (x^3 + 1) - e^2 \cdot 3x^2}{(x^3 + 1)^2} = \frac{-3e^2 x^2}{(x^3 + 1)^2} $. ‖ ∻Answer∻ ⚹ $ f'(x) = \frac{-3e^2 x^2}{(x^3 + 1)^2} $ ⚹ ∻Key Concept∻ ⚹Quotient Rule for Derivatives⚹ ∻Explanation∻ ⚹The quotient rule is used to differentiate functions that are the ratio of two other functions, allowing us to find the derivative accurately.⚹◊What is the Quotient Rule for differentiation?⍭ Generate me a similar question◊

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